I would definitely involve your insurance company and let them know about the situation. The other driver's insurance company should be paying for your damage up to the limit of that policy, but if it doesn't cover everything then your current company should be able to help cover the rest.
Accidents that occur on private property do not require a police report in order to file a claim. Your familiarity with the other driver does not impact your ability to file a claim, however the fact that he is uninsured means that you will have to pay your deductible for the repairs.
If the other party hit your vehicle, and it's pretty clearly they're at fault, then you should contact their insurance for repairs. Parking lot accidents or accidents that occur on private property can be difficult to determine fault, but it sounds like the incident is pretty straight forward.
A golf cart is covered on a Special Lines insurance policy since it is a different type of risk from a traditional automobile. Unfortunately that means you won't be able to add it with your vehicle, but most insurance companies offer discounted rates for "bundling" your two policies.